Question: Solve for $x$ : $ 2|x + 7| + 9 = 4|x + 7| + 3 $
Answer: Subtract $ {2|x + 7|} $ from both sides: $ \begin{eqnarray} 2|x + 7| + 9 &=& 4|x + 7| + 3 \\ \\ {- 2|x + 7|} && {- 2|x + 7|} \\ \\ 9 &=& 2|x + 7| + 3 \end{eqnarray} $ Subtract $3$ from both sides: $ \begin{eqnarray} 9 &=& 2|x + 7| + 3 \\ \\ {- 3} && {- 3} \\ \\ 6 &=& 2|x + 7| \end{eqnarray} $ Divide both sides by ${2}$ $ \dfrac{6} {{2}} = \dfrac{2|x + 7|} {{2}} $ Simplify: $ 3 = |x + 7| $ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ -3 = x + 7 $ or $ 3 = x + 7 $ Solve for the solution where $x + 7$ is negative: $ - 3 = x + 7$ Subtract ${7}$ from both sides: $ \begin{eqnarray} - 3 &=& x + 7 \\ \\ {- 7} && {- 7} \\ \\ -3 - 7 &=& x \end{eqnarray} $ $ -10 = x $ Then calculate the solution where $x + 7$ is positive: $ 3 = x + 7 $ Subtract ${7}$ from both sides: $ \begin{eqnarray} 3 &=& x + 7 \\ \\ {- 7} && {- 7} \\ \\ 3 - 7 &=& x \end{eqnarray} $ $ -4 = x $ Thus, the correct answer is $x = -10 $ or $x = -4 $.